Viewing and Projections · A few more steps expanded Application Vertex batching & assembly...
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University of Texas at Austin CS354 - Computer Graphics Don Fussell
Viewing and Projections
Don Fussell Computer Science Department
The University of Texas at Austin
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A Simplified Graphics Pipeline Application
Vertex batching & assembly
Triangle assembly
Triangle clipping
Triangle rasterization
Fragment shading
Depth testing
Color update Framebuffer
NDC to window space
Depth buffer
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A few more steps expanded Application
Vertex batching & assembly
Lighting
View frustum clipping
Triangle rasterization
Fragment shading
Depth testing
Color update Framebuffer
NDC to window space
Depth buffer
Vertex transformation
User defined clipping
Back face culling
Perspective divide
Triangle assembly Texture coordinate generation
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Conceptual Vertex Transformation glVertex* API commands
Modelview matrix
User-defined clip planes
View-frustum clip planes
to primitive rasterization
object-space coordinates (xo,yo,zo,wo)
eye-space coordinates (xe,ye,ze,we)
clipped eye-space coordinates
clipped clip-space coordinates Perspective
division Projection matrix
Viewport + Depth Range transformation
(xc,yc,zc,wc)
window-space coordinates
(xw,yw,zw,1/wc)
normalized device coordinates (NDC)
(xn,yn,zn,1/wc)
clip-space coordinates
(xc,yc,zc,wc)
(xe,ye,ze,we)
(xe,ye,ze,we)
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Eye Coordinates (not NDC)
-Z direction “looking into the screen”
+X -X
+Y
-Y
+Z direction “poking out of the screen”
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Planar Geometric Projections
Standard projections project onto a plane Projectors are lines that either
converge at a center of projection are parallel
Such projections preserve lines but not necessarily angles
Nonplanar projections are needed for applications such as map construction
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Classical Projections
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Perspective vs Parallel
Computer graphics treats all projections the same and implements them with a single pipeline Classical viewing developed different techniques for drawing each type of projection Fundamental distinction is between parallel and perspective viewing even though mathematically parallel viewing is the limit of perspective viewing
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Taxonomy of Projections
parallel perspective
axonometric multiview orthographic
oblique
isometric dimetric trimetric
2 point 1 point 3 point
planar geometric projections
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Parallel Projection
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Perspective Projection
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Orthographic Projection
Projectors are orthogonal to projection surface
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Multiview Orthographic Projection
Projection plane parallel to principal face Usually form front, top, side views
isometric (not multiview orthographic view) front
side top
in CAD and architecture, we often display three multiviews plus isometric
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Advantages and Disadvantages
Preserves both distances and angles Shapes preserved Can be used for measurements Building plans Manuals
Cannot see what object really looks like because many surfaces hidden from view
Often we add the isometric
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Projections and Normalization
The default projection in the eye (camera) frame is orthogonal For points within the default view volume
Most graphics systems use view normalization All other views are converted to the default view by transformations that determine the projection matrix Allows use of the same pipeline for all views
xp = x yp = y zp = 0
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Default Projection
Default projection is orthographic
clipped out
z=0
2
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Orthogonal Normalization
glOrtho(left,right,bottom,top,near,far)
normalization ⇒ find transformation to convert specified clipping volume to default
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OpenGL Orthogonal Viewing
glOrtho(left,right,bottom,top,near,far)
near and far measured from camera
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Homogeneous Representation
xp = x yp = y zp = 0 wp = 1
pp = Mp
M =
!!!!
"
#
$$$$
%
&
1000000000100001
In practice, we can let M = I and set the z term to zero later
default orthographic projection
University of Texas at Austin CS354 - Computer Graphics Don Fussell
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Orthographic Eye to NDC Two steps
Move center to origin T(-(left+right)/2, -(bottom+top)/2,-(near+far)/2))
Scale to have sides of length 2 S(2/(left-right),2/(top-bottom),2/(near-far))
2right − left
0 0 −right + leftright − left
0 2top− bottom
0 −top+ bottomtop− bottom
0 0 2near − far
−far + nearfar − near
0 0 0 1
"
#
$$$$$$$$$
%
&
'''''''''
P = ST =
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University of Texas at Austin CS354 - Computer Graphics Don Fussell
Orthographic Transform Prototype
glOrtho(GLdouble left, GLdouble right, GLdouble bottom, GLdouble top, GLdouble near, GLdouble far)
Post-concatenates an orthographic matrix
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
+−
−
−−
+−
−
−
+−
−
1000
200
020
002
nfnf
nf
btbt
bt
lrlr
lr
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glOrtho Example
Consider glLoadIdentity(); glOrtho(-20, 30, 10, 60, 15, -25)
left=-20, right=30, bottom=10, top=50, near=15, far=-25
Matrix
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
+−
−
−−
+−
−
−
+−
−
1000
200
020
002
nfnf
nf
btbt
bt
lrlr
lr
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
−
100041
20100
230
2010
5100
251
=
-Z axis
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Axonometric Projections
Allow projection plane to move relative to object
classify by how many angles of a corner of a projected cube are the same none: trimetric two: dimetric three: isometric
θ 1
θ 3 θ 2
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Types of Axonometric Projections
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Advantages and Disadvantages
Lines are scaled (foreshortened) but can find scaling factors Lines preserved but angles are not
Projection of a circle in a plane not parallel to the projection plane is an ellipse
Can see three principal faces of a box-like object Some optical illusions possible
Parallel lines appear to diverge Does not look real because far objects are scaled the same as near objects Used in CAD applications
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Oblique Projection Arbitrary relationship between projectors and
projection plane
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Advantages and Disadvantages
Can pick the angles to emphasize a particular face Architecture: plan oblique, elevation oblique
Angles in faces parallel to projection plane are preserved while we can still see “around” side
In physical world, cannot create with simple camera; possible with bellows camera or special lens (architectural)
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Perspective Projection
Projectors coverge at center of projection
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Vanishing Points
Parallel lines (not parallel to the projection plan) on the object converge at a single point in the projection (the vanishing point) Drawing simple perspectives by hand uses these vanishing point(s)
vanishing point
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Three-Point Perspective
No principal face parallel to projection plane Three vanishing points for cube
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Two-Point Perspective
On principal direction parallel to projection plane Two vanishing points for cube
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One-Point Perspective
One principal face parallel to projection plane One vanishing point for cube
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33 Perspective in Art History
[Pietro Perugino, 1482]
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34 Perspective in Art History
[Pietro Perugino, 1482]
Vanishing point
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35 Humanist Analysis of Perspective
[Albrecht Dürer, 1471]
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Advantages and Disadvantages
Objects further from viewer are projected smaller than the same sized objects closer to the viewer (diminution)
Looks realistic Equal distances along a line are not projected into equal distances (nonuniform foreshortening) Angles preserved only in planes parallel to the projection plane More difficult to construct by hand than parallel projections (but not more difficult by computer)
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1-, 2-, and 3-point Perspective A 4x4 matrix can represent 1, 2, or 3 vanishing points
As well as zero for orthographic views
3-point perspective
2-point perspective 1-point perspective
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Simple Perspective
Center of projection at the origin Projection plane z = d, d < 0
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Perspective Equations
Consider top and side views
xp =
dzx/
dzx/
yp = dzy/
zp = d
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Homogeneous Form
consider q = Mp where M =
!!!!
"
#
$$$$
%
&
0/100010000100001
d
!!!!
"
#
$$$$
%
&
1zyx
!!!!
"
#
$$$$
%
&
dzzyx
/
q = ⇒ p =
University of Texas at Austin CS354 - Computer Graphics Don Fussell
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OpenGL Perspective
glFrustum(left,right,bottom,top,near,far)
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Simple Perspective
Consider a simple perspective with the COP at the origin, the near clipping plane at z = -1, and a 90 degree field of view determined by the planes
x = ±z, y = ±z
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Simple Eye to NDC
N =
1 0 0 00 1 0 00 0 α β
0 0 −1 0
"
#
$$$$
%
&
''''
after perspective division, the point (x, y, z, 1) goes to
x’ = x/z y’ = y/z z’ = -(α+β/z)
which projects orthogonally to the desired point regardless of α and β
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Picking α and βIf we pick
α =
β =
nearfarfarnear
−
+
farnearfarnear2
−
∗
the near plane is mapped to z = -1 the far plane is mapped to z =1 and the sides are mapped to x = ± 1, y = ± 1
If we start from the simple eye frustum, we end up with the NDC clipping cube
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Normalization Transformation
original clipping volume original object new clipping
volume
distorted object projects correctly
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University of Texas at Austin CS354 - Computer Graphics Don Fussell
Frustum Transform Prototype
glFrustum(GLdouble left, GLdouble right, GLdouble bottom, GLdouble top, GLdouble near, GLdouble far)
Post-concatenates a frustum matrix
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−
−
+−−
+
−
−
+
−
0100
2)(00
020
002
nffn
nfnfbtbt
btn
lrlr
lrn
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glFrustum Matrix
Projection specification glLoadIdentity(); glFrustum(-4, +4, -3, +3, 5, 80) left=-4, right=4, bottom=-3, top=3, near=5, far=80
Matrix
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−
010075800
758500
00350
00045
=
-Z axis
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−
−
+−−
+
−
−
+
−
0100
2)(00
020
002
nffn
nfnfbtbt
btn
lrlr
lrn
symmetric left/right & top/bottom so zero
80 5
![Page 48: Viewing and Projections · A few more steps expanded Application Vertex batching & assembly Lighting View frustum clipping ... 1 point 2 point 3 point planar geometric projections](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facd33e83ce8d67f109edab/html5/thumbnails/48.jpg)
University of Texas at Austin CS354 - Computer Graphics Don Fussell
48 glFrustum Example
Consider glLoadIdentity(); glFrustum(-30, 30, -20, 20, 1, 1000)
left=-30, right=30, bottom=-20, top=20, near=1, far=1000
Matrix
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−
01009992000
999100100
002010
000301
=
-Z axis
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−
−
+−−
+
−
−
+
−
0100
2)(00
020
002
nffn
nfnfbtbt
btn
lrlr
lrn
symmetric left/right & top/bottom so zero
![Page 49: Viewing and Projections · A few more steps expanded Application Vertex batching & assembly Lighting View frustum clipping ... 1 point 2 point 3 point planar geometric projections](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facd33e83ce8d67f109edab/html5/thumbnails/49.jpg)
University of Texas at Austin CS354 - Computer Graphics Don Fussell
glOrtho and glFrustum
These OpenGL commands provide a parameterized transform mapping eye space into the “clip cube” Each command
glOrtho is orthographic glFrustum is single-point perspective
![Page 50: Viewing and Projections · A few more steps expanded Application Vertex batching & assembly Lighting View frustum clipping ... 1 point 2 point 3 point planar geometric projections](https://reader034.fdocuments.net/reader034/viewer/2022050605/5facd33e83ce8d67f109edab/html5/thumbnails/50.jpg)
Next Lecture More viewing Transform from object to eye space
University of Texas at Austin CS354 - Computer Graphics Don Fussell